Coarse selectors of graphs

We consider a connected graph Γ as a coarse space and prove that Γ admits a 2-selector if and only if Γ is either bounded or coarsely equivalent to \({\mathbb{N}}\) $\mathbb{N}$ or \({\mathbb{R}}\) $\mathbb{R}$ . We apply this result to geodesic metric spaces admitting the linear orders compatible with coarse structures.